Actually, there was multiple HMS Enterprize's and HMS Enterprise's. Around the period of 1775 there was a HMS Enterprise that was captured by the Americans but the ships were typically called "Enterprize" up until WWII. For example there is an Echo-class survey ship in service with the Royal Navy at this current time calle HMS Enterprise. And yes the ship shown at the start of ENT is HMS Enterprize so therefore the ship on the wall should read Enterprize.

EDIT: The ship captured in 1775 by the Americans was HMS Enterprize not Enterprise

And there was no ship launced under the name Enterprise or Enterprize in 1775